This reading introduced the idea of flows on the circle. The circle is one-dimensional, like the line, but now periodic solutions are possible. If we let o=theta, the equation looks like odot=f(o) where o is a point on the circle and odot is its velocity. Vector fields are sketched in a similar way. If odot=w, where w is a constant then there is uniform motion around the circle and the solution is periodic with the period, T, equal to 2pi/w. A nonuniform oscillator like odot=w-a(sin(o)) has a saddle-node bifurcation and a "bottleneck" near o=pi/2. A bottleneck is an area where it takes the phase point most of its time to pass through. In fact, when calculating periods, we can approximate the entire time as just the time spent in the bottleneck.
Challenges
Ghosts. The idea seems cool, but I don't quite follow why ghosts exist and how you determine the time spent in a bottleneck because of a ghost.
Reflections
I like the fact that equations like odot=sin(o) no longer have infinite fixed points, which I found to be a hassle. But so far it is slightly harder to visualize flows and vector fields in a circular manor.
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